# Strongly minimal edge cover

For a hypergraph $H=(V,E)$, we call $C\subseteq E$ an edge cover if $\bigcup C =V$. An edge cover $C$ is strongly minimal if $\left|C\setminus C'\right|\leq \left|C'\setminus C\right|$ holds for any edge cover $C'$.
Is it true that if the hypergraph $H$ has no isolated vertices and all of its hyperedges are finite, then $H$ admits a strongly minimal edge cover?